Im asked to solve a PDE $2u_x+3u_y = xu+x$ where $u(0,y) = y$.
I tried to solve the problem with https://en.wikipedia.org/wiki/Method_of_characteristics.
Hence my parametrization is $\frac{dx}{2} = \frac{dy}{3} = \frac{du}{xu(x,y)+x}$. Then $c_1 = y-\frac{3}{2}x$ and then i need to solve an ODE $\frac{du}{dx} = \frac{xu(x,y)+x}{2}$ which have a solution $c_1e^{\frac{x^2}{4}} - 1$.
Then clearly the solution is not correct when substituting $c_1$ to the 2nd solution hence $(y-\frac{3}{2}x)e^{\frac{x^2}{4}} - 1$ . How to deal with the initial value condition when solving the latest ODE?